Next-to-next-to-leading order matching of beauty-charmed meson $B_c$ and $B_c^{\ast }$ decay constants

We present the next-to-next-to-leading order (NNLO) QCD corrections to the decay constants for both the pseudoscalar and vector beauty-charmed mesons $B_c$ and $B_c^{\ast }$ in nonrelativistic QCD effective theory. Explicit NNLO calculation verified that the $B_c$ decay constant from pseudoscalar current is identical with the $B_c$ decay constant from axial-vector current. The NNLO result for the vector decay constant of $B_c^{\ast }$ meson is novel. Combined with the latest extraction of nonrelativistic QCD long-distance matrix elements of $B_c$ meson, we give the branching ratios of leptonic decays of $B_c$ and $B_c^{\ast }$ mesons. In addition, the novel anomalous dimension for the flavor-changing heavy quark vector current in nonrelativistic QCD effective theory are helpful to investigate the threshold behaviours of two different heavy quarks.